Quantum chemistry and wavefunction based methods for ... · 7/12/2011 · The 1998 Nobel Prize in...

HUMBOLDT-UNIVERSITÄT ZU BERLIN

Quantum chemistry and wavefunction basedmethods for electron correlation

Hands-on Tutorial Workshop „Ab initio Molecular Simulations“Fritz-Haber-Institute, Berlin, July 12, 2011

Joachim SauerInstitut für Chemie, Humboldt-Universität

©Joachim Sauer, HU Berlin, 2011

The Royal Swedish Academy of Sciences has awarded The 1998 Nobel Prize in Chemistry in the area of quantum chemistry to

Walter Kohn, University of California at Santa Barbara, USA andJohn A. Pople, Northwestern Univ., Evanston, Ill., USA (British citizen).

Citation:"to Walter Kohn for his development of the density-functional theory and toJohn Pople for his development of computational methods in quantumchemistry."


Surface Reaction

TS

Catalyst+ Educt

IntrinsicBarrier

Catalyst-EductComplex

Catalyst-ProductComplex

Bindingenergy

Reaction energy

ApparentBarrier


DFT problems with barriers

TS

Catalyst+ Educt

IntrinsicBarrier

Catalyst-EductComplex

Catalyst-ProductComplex

Bindingenergy

Van der Waals(dispersion)

SI error(reaction site)


Zeolite catalysts: active sites (transition metal ions, protons)in a „surface-only“ silica matrix

H

-Si+Al,

AlSiO O

O O

AlSiO O

O O

H

SiSiO O

O OM+

H

-Si+Al,

AlSiO O

O O

AlSiO O

O O

H

SiSiO O

O OM+


Zeolite catalysis: Methanol-to-hydrocarbons

Review: Stöcker, Microp. and Mesop. Mat. 29 (1999) 3-48

Methanol fromdifferent sources

CH3OH CH3OCH3

CH2=CH2

CH2=CH-CH3↓

Cyclic Polyens

Aromatic HC

Methylation

C-C Formation, induction

↓

↓

↓

+ CH3OH

GasolineOlefines

MTGMTO

Hydrocarbon poolmechanism

Haw et al, Kolboe et al.


MeOH (g)Alkene (g)

MeOH (ads)Alkene (g)

MeOH (ads)Alkene (ads) Water (ads)

Alkene (ads)

Water (g)Alkene (g)

Transitionstructure

*S. Svelle, P.A. Ronning, S. Kolboe, J. Catal. 2004, 224, 115.S. Svelle, P.O. Ronning, U. Olsbye, S. Kolboe,J. Catal. 2005, 234, 385.

Methylation of ethene, propene, trans-2-butene in H-MFI

Measured* Apparent BarrierEthene 104Propene 64 (-40)t-2-Butene 40 (-24)


Methylation of alkenes in H-MFI - pbc DFT

Svelle, Tuma, Rozanska, Kerber, Sauer, JACS 131 (2009) 816

Error increases with chain length


s6 = 1.4/1.3/0.7 for BLYP/BP86/PBE

Parameters are as good for solids (condensed systems) as formoleculesNote, however, Mg2+ very different from Mg, whereas O2- similar to O

Implementation of Ewald sum for 1/r6 for periodic systems

Etotal = EDFT +Edisp

Edisp = !s6C6

ij

rij6j = i +1

N

"i =1

N !1

" fdamp fdamp (R) = 1

1+e-! (R/R0 -1)

Pragmatic solution: DFT + Dispersion (DFT+D)

Grimme, J. Comput. Chem., 2004, 25, 1463; 2006, 27, 1787.

Many predecessors with DFT, HF+Disp, e.g. Ahlrichs and ScolesMany more sophisticated schemes

Kerber, Sierka, Sauer, J. Comput. Chem. 29 (2008) 2088.


Methylation of alkenes in H-MFI - pbc DFT+D


Dispersion - increases with chain length


Methylation of alkenes in H-MFI - pbc DFT+D


Dispersion

Too low barrierconstant SIC error


Errors on energy barriers (Truhlar et al.)

Nucleophilic substitution

kcal/mol

Reference data:CCSD(T) with extrapolation to complete basis set limit(„W1 theory“)


DFT (PBE/plane waves), VASP20.2x20.5x13.5 Å; 96 +18 atomsPeriodic Boundary Conditions

b _

_a

CCSD(T)/cbs17+17 AtomeC3O11Si2AlH17MOLPRO

MP2/TZVP123+67AtomeC3O72Si47AlH67CC2 code

Divide and Conquer - Models and Methodsn


Hybrid high level : low level method

Ehybrid (S,C) = EDFT+D(S) + [EMP2(C) - EDFT+D(C)]

High level correction

Step 0: PBE+D optimization, periodic boundary conditions (pbc) Frequency calculation for stationary points, ZPVE

Step 1: Hybrid MP2(cluster):PBE+D(pbc) optimization[Step 2: Basis set extrapolation to CBS limit, single point Step 3: CCSD(T)-MP2, small cluster model

Tuma, Sauer, CPL 2004, 387, 388; PCCP, 2006, 8, 3955Kerber, Sierka, Sauer, J. Comput. Chem. 2008, 29, 2088Reuter/Scheffler, PRL 98 (2007) 176103 (CO/Cu(111))

Stoll, JPC A 113 (2009) 11483 (Be, Mg crystals)

Hybrid MP2(cluster):PBE+D(pbc) +ΔCCSD(T) method

© Joachim Sauer, HU Berlin, 2011

Energy as functional of orbitals or electron density

kinet. e-n e-e e-e en. attract Coulomb exchange/corr orbital density density

EHF = ET + EN + EJ + EXFock30 (orbital)

EDFT = ET + EN + EJ + EXC (density)functional of density onlyEXC

LDA = EXDirac30 + EC

VWN

functional also of density gradient (Generalized Gradient Approximation)EXC

BLYP = EXDirac30 + ΔEX

B88 + ECLYP

Self-interactioncancels


12 !(x1) 1r12"" !(x2 )dx1dx2 # 1

2 i*(x1) j$(x2 )"" 1

r12j (x1)i (x2 )dx1dx2

i, j%!(x) =

i(x)i=1

N

" i*(x)

Coulomb and exchange terms cancel - self-interaction correction (SIC)

Self-interaction correctionHartree-Fock EJ + EX

F30 (orbital)

12 ij ij!"i, j# $ ij ji %&

i = j

Kohn-Sham EJ + EXC (density)12 !(x1) 1r12"" !(x2 )dx1dx2 + dx!(x)VXC !,#![ ]"

12 ii ii + i VXC !,"![ ] i # 0

Coulomb and exchange do not cancel - self-interaction error

* i*(x1) j!(x2 )"" 1

r12i (x1) j (x2 )dx1dx2 = ij ij

ii ii!" # ii ii $% = 0


kinet. e-n e-e e-e en. attract Coulomb exchange/corr orbital density density

EHF = ET + EN + EJ + EXFock30 (orbital)

EDFT = ET + EN + EJ + EXC (density)functional of density onlyEXC

LDA = EXD30 + EC

VWN

functional also of density gradient (Generalized Gradient Approximation)EXC

BLYP = EXDirac30 + ΔEX

B88 + ECLYP

orbital-density hybrid functionalEXC

B3LYP = a0 EXF30 + (1- a0) EX

D30 + aX ΔEXB88

+(1- aC ) ECVWN + aC EC

LYP

Energy as functional of orbitals or electron density

Self-interactioncancels

Self-interactioncancels partially


SOMO

20% 0% 50% Fock-exchange in functional

Cluster anions (V2O5)n-

Size-dependentelectron localization


V6O15-

Cs- 2Aʻ-10

C2v- 2A2- 9

- 48 kJ/mol

BH-LYPCCSD(T)//BH-LYPB3-LYP

- 57 kJ/mol

Cs- 2Aʻʻ

D2d- 2B1

Cs- 2Aʻʻ

V4O10-

+ 33

CCSD(T) calculations confirm most stable structures


Orbital functional for electron correlation:2nd order Møller-Plesset perturbation theory

ECMP2 ![ ] = "

ab rs 2

#r + # s " #a " #br<s$

a<b$

Covers dispersionFunctionals including this term and Fock exchange: double hybrid

EDFT = ET + EN + EJ + EXC [ρ] + ax EXF30 + aMP2 EC

MP2


! = !o + Cia!i

a

i,a" + Cij

ab!ijab

i< j ,a<b

"

+ Cijkabc!ijk

abc +…i< j<k ,a<b<c

"

Wave function expansion


Wave function expansion

! = Cn!nn" = !o + Ci

a!ia

i,a" + Cij

ab!ijab

i< j ,a<b

" + Cijkabc!ijk


"

Result depends on- configuration space „full configuration interaction“- size of one-particle space (M, given by the basis set)

how many configurations of a given substitution level ?

M !N !(M ! N )!

= 4 "1011 for CH4 DZP basis M=70, N=10


M-N (virtual space)

pVDZ


HF/small basis set

HF/large basis set

HF-limit

El.corr.calculated

Accurate, non-rel.

Exact (=exp.)

El. correlationsenergy (Def.)

El. corr. energyin quantum chemical

practise

Energy


Electron correlation - „Post Hartree Fock“

! = !o + Cia!i

a

i,a" + Cij

ab!ijab

i< j ,a<b

" + Cijkabc!ijk


"

Ecorr = E ! EHF = Cabrs "0

r<sa<b

# H "abrs = Cab

rs

r<sa<b

# ab rs

! a*(x1)! b

"(x2 )# 1r12! r (x1)! s (x2 )dx1dx2 = ab rsab rs = ab rs ! ab sr

You only need double substitution to calculate the exact correlation energy,but to determine the doubles coefficients exactly, you need all substitutions


Many body perturbation theory (MBPT)

H0 = F(! )!"

#0 $ - Determinant consisting of canonical HF orbitals

E(0) = %ii" E (0) + E (1) = EHF

Ecorr = E(2) = !

ab rs"r + " s ! "a ! "br<s

#a<b# ab rs = EMP2

!0 H !ar = 0 (BrillouinTheorem)

!0 V !abrs = !0 H !ab

rs = ab rs = cµac"bc# rc$sµ"#$% µ" #$

EXCMP2 = EX

F30 + ECMP2


Limiting value

Typical oscillating behaviour of results obtained with the MP method

Energy (Property)

SCF

MP3

MP4MP2


Coupled Cluster Expansion

! = eT"0 = (1+ T + 12 T2 + 1

3! T3 + ....)"0

eT =1+ T1 + (T2 + 12 T1

2) + (T3 + T2T1 + 13! T1

3) + (T4 + 12 T2

2 +

T1!0 =i" ti

m

m" !i

m

T12!0 =

i,j" ti

mtjn

m,n" ! i j

mn

T2!0 = t i jmn

m,n" ! i j

mn

i,j"

disconnected (unlinked) doubles

connected (linked) doubles

t - Amplitudes

The presence of disconnected (double) substitutionensures size-consistency


Coupled Cluster Energy

Ecorr =i < j! (t ijmn + timt jn "

m<n! tint jm ) ij mn

Equations for determination of „amplitudes“ (coefficients)

(if HF orbitals are used)

timim! "l

p H"im + (t ijmn + timtjn # tintjm)

ij,mn! "l

p H" ijmn +

(t ijmntko + ...timtjntko # ...)! "lp H" ijk

mno = Etlp

Iterative solution

E= !0 H 1+T1+(T2 + 12 T1

2 )!0


CC-SD equation (HF-Orbitale)

!im H (T1 + T2 + 1

2 T12 + T1T2 + 1

6 T13 )!0 = E " ti

m

! ijmn H (1+ T1 + T2 + 1

2 T12 + T1T2 + 1

6 T13 + 1

2 T22 + 1

2 T12T2 + 1

24 T14 )!0

= E " (t ijmn + timtjn # tintjm)

CCSD(T) - (T)riple-contribution according to MP4 equation,but T1 and T2 amplitudes are taken from CCSD equations


Wavefunction-based methods for electron correlation

I. According to the way, the expansion coefficients are determinedvariational: Configuration Interaction (CI methods)

Size-consistency problemCIS (CI with singles) for excited states

perturbation theory: Møller-PlessetMP2, MP4-SDTQ

system of equations: Coupled Cluster (iterative)CCSD; CCSD(T)


II. Substitutions included in wavefunction expansionall in given one-particle basis: full CIup to quadruple: MP4-SDTQup to connected triple substitution: CCSD(T)

(an expensive standard method)up to doubles: CI-SD → CPFdoubles only: MP2

(the simplest wavefunction-based standard method))singles only: CI-S

(simplest method for excited states)

Wavefunction-based methods for electron correlation


Neese et al., Acc. Chem. Res. 42 (2009) 641-648


The scaling problem


Pople school

(GAUSSIAN)

correlation consistent

Dunning

6-311G**

6-311+G(2df,p)

6-311+G(3df,2p)

cc-pVDZ

cc-pVTZ

cc-pVQZ

3s2p1d/2s1p

4s3p2d1f/3s2p1d

5s4p3d2f1g/4s3p2d1f

+ diffuse s-function

aug - diffuse functions (one for each

angular momentum)

Polarisation functions indispensable/diffuse functions recommended

Electron correlation: requires better (larger) basis setsthan Hartree Fock or DFT


Dunning (aug)-cc-pVnZ Basis Sets

s p d f g h No.AO X/H

cc-pVDZ 3/2 2/1 1/0 14/05cc-pVTZ 4/3 3/2 2/1 1/0 30/14cc-pVQZ 5/4 4/3 3/2 2/1 1/0 55/30 cc-pV5Z 6/5 5/4 4/3 3/2 2/1 1/0 91/55

aug-cc-pVDZ 4/3 3/2 2/0 23/09aug-cc-pVTZ 5/4 4/3 3/2 2/0 46/23aug-cc-pVQZ 6/5 5/4 4/3 3/2 2/0 80/46aug-cc-pV5Z 7/6 6/5 5/4 4/3 3/2 2/0 127/80

l/k refers to first row atom/h atom


How to reach the basis set limitfor a given method?

• Use basis sets of increasing size• Reaching the basis set limit faces the scaling problem• More expensive methods generally need larger basis sets

to reach limit• Extrapolation formulas available


Basis set extrapolation - CBS limit

(aug)-cc-pVXZ

Two-point extrapolationA. Halkier et al.,CPL 286 (1988) 243

other and more sophisticated schemes, see refs. inT. H. Dunning, Jr., JPC A 104 (2000) 9062

Note: You need to extrapolate also the Hartree-Fock energy


O2 dissociation energy (kJ/mol)

Method Basis set De D0 Obsd 505 494 UCCSD(T)FC cc-pVTZ 476

cc-pVQZ 490 CBS(T,Q) 501 aug-cc-pVTZ 481 CBS 501 CBS+cv 503

G2 G2 494 484

EMSL Computational Results DataBase - 255 Molecules, 41 Atomshttp://www.emsl.pnl.gov:2080/proj/crdb/


Au-S bond distance (pm) and H2S-Au interaction energies

(kJ/mol)

RAu-S E

PBE/TZVPP 247 -53.8

PBE+Db/TZVPP//PBE 247 -56.9

B3LYP/TZVPP//B3LYP 260 -26.5

B3LYP+Db/TZVPP 260 -30.7

CCSD(T)//B3LYP+Da E E/CBS(X,Y)

aug-pVDZ -27.4 -

aug-pVTZ -32.8 -34.7 (D,T)

aug-pVQZ -35.4 -37.0 (T,Q)

aug-pV5Z -36.7 -37.7 (Q,5)

b Au C6 parameters from Tonigold and Gross ON TOPON TOPFLATFLATAuAu

Au_adAu_adSS

GAS PHASEGAS PHASE

H HS

Au


Basis Set Superposition Error

ΔE = EAB - ( EA + EB )

ΔEc = EAB - ( EA(B) + E(A)B )

εA = (EA - EA(B)) εB = (EB - E(A)B)

Boys Bernardi function counterpoise method


BSSE

BSSEBasis set superposition errorBSCEBasis set convergence error

Thom H. Dunning Jr."A road map for the calculationof molecular binding energies"J Phys Chem A 2000, 104, 9062-9080


Notes on BSSE

• BSSE for two subsystems is an indicator of balanced basis sets• For small basis sets BSSE correction may lead to larger

deviations from experiment (or basis set limit)• BSSE corrected results are less basis set dependent• There may be other errors (BSCE)• Sizeable for intermolecular complexes, but also present for

chemical bonds• BSSE elimination is not always easy. How do you correct an

energy barrier for BSSE?• There is no BSSE for plane wave basis sets


There are cases, for which the Hartree Fock determinant is not asufficiently good starting pointStretched, almost dissociated bonds (H2 example)Biradicals, polynuclear transition metal compoundsAlmost degenerate states (partially occupied d-shells)

Multireference casesCAS (Complete Actice Space) SCF

MC (Multi-configuration) SCF+ MR (Multireference)-CI(SD)


GGA functionals (PBE) are often not good enough, buthybrid functionals (B3LYP, PBE0, HSE) are one order of magnitude moreexpensive (with plane waves and pbc two orders of magnitude)

GGA functionals yield more delocalized electronic structures(delocalization of holes), preference for polar structures and too lowenergy barriers.

Partially occupied d-shells, the presence of more than one TM site andthe decoupling of electron pairs in bond breaking processes requires amulti-reference treatment- DFT limited to high-spin, low spin states require broken symmetryapproximation- in general, CCSD(T) is not good enough and not suitable as benchmark- ab initio multi-reference methods cannot treat large enough systemsreliably.

The challenge of transition metal oxides


T1 diagnostics

Is CCSD(T) applicable?Is a single determinant a good reference function?

T1 > 0.02 not a clear single reference state

T1!0 =i" ti

m

m" !i

m

T1 = t1 / Ne t1 =i! (ti

m

m! )2

!= 1[ +T1+(T2 + 12T1

2 )+ (T2T1 + 13!T1

3 )+ ...]"0


V2O5/SiO2 OV(OCH3) OV(OCH3)+

Calculated broken symmetry doublet

E#,B3LYP 154 147 80E#,CCSD(T) (137) 131ObservedE#

intrinsic (0 K) 137±10 52±11estimate

VV(d0)VIV(d1) - biradicaloid

broken symmetry VIII(d2)- triplet

H-Transfer: surface complex - gas phase model

Döbler, Pritzsche, Sauer, JACS 127 (2005) 10861

T1=0.07


If you would like to join our group aspostdoc or PhD student -

[email protected]

(www.chemie.hu-berlin.de/ag_sauer

Quantum chemistry and wavefunction based methods for ... · 7/12/2011 · The 1998 Nobel Prize in...

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Transcript of Quantum chemistry and wavefunction based methods for ... · 7/12/2011 · The 1998 Nobel Prize in...